The Dicke Quantum Phase Transition with a Superfluid Gas in an Optical Cavity
Kristian Baumann, Christine Guerlin, Ferdinand Brennecke, Tilman Esslinger
TThe Dicke Quantum Phase Transition with a Superfluid Gas in an Optical Cavity
Kristian Baumann, Christine Guerlin, ∗ Ferdinand Brennecke, and Tilman Esslinger † Institute for Quantum Electronics, ETH Z¨urich, CH–8093 Z¨urich, Switzerland (Dated: May 28, 2018)A phase transition describes the sudden change of state in a physical system, such as the transitionbetween a fluid and a solid. Quantum gases provide the opportunity to establish a direct link betweenexperiment and generic models which capture the underlying physics. A fundamental concept todescribe the collective matter-light interaction is the Dicke model which has been predicted to showan intriguing quantum phase transition. Here we realize the Dicke quantum phase transition inan open system formed by a Bose-Einstein condensate coupled to an optical cavity, and observethe emergence of a self-organized supersolid phase. The phase transition is driven by infinitelylong-range interactions between the condensed atoms. These are induced by two-photon processesinvolving the cavity mode and a pump field. We show that the phase transition is described bythe Dicke Hamiltonian, including counter-rotating coupling terms, and that the supersolid phase isassociated with a spontaneously broken spatial symmetry. The boundary of the phase transition ismapped out in quantitative agreement with the Dicke model. The work opens the field of quantumgases with long-range interactions, and provides access to novel quantum phases.
INTRODUCTION
The realization of Bose-Einstein condensation (BEC)in a dilute atomic gas marked the beginning of a newapproach to quantum many-body physics. Meanwhile,quantum degenerate atoms are regarded as an ideal toolto study many-body quantum systems in a very wellcontrolled way. Excellent examples are the BEC-BCScrossover and the observation of the superfluid toMott-insulator transition . The high control availableover these many-body systems has also stimulated thenotion of quantum simulation , one of the goals being togenerate a phase diagram of an underlying Hamiltonian.However, the phase transitions and crossovers which havebeen experimentally investigated with quantum gases upto now are conceptually similar since their physics is gov-erned by short-range interactions.In order to create many-body phases dominated bylong-range interactions different routes have been sug-gested, most of which exploit dipolar forces betweenatoms and molecules . A rather unique approach con-siders atoms inside a high-finesse optical cavity, so thatthe cavity field mediates infinitely long-range forces be-tween all atoms . In such a setting a phase transi-tion from a Bose-Einstein condensate to a self-organizedphase has been predicted once the atoms induce a suf-ficiently strong coupling between a pump field and anempty cavity mode . Indeed, self-organization ofa classical, laser-cooled atomic gas in an optical cav-ity was observed experimentally . Conceptually re-lated experiments studied the atom-induced couplingbetween a pump field and a vacuum mode using ul-tracold or condensed atoms. This led to the observa-tion of free-space and cavity-enhanced superradi-ant Rayleigh scattering, as well as to collective atomic re-coil lasing . Both phenomena did not support steady-state quantum phases, and became visible in transientmatter wave pulses.A rather general objective of many-body physics is to understand quantum phase transitions and to un-ravel their connection to entanglement . An impor-tant concept within this effort is a system of interactingspins in which each element is coupled to all others withequal strength. The most famous example for such an in-finitely coordinated spin system is the Dicke model ,which has been predicted to exhibit a quantum phasetransition more than thirty years ago . The Dickemodel considers an ensemble of two-level atoms, i.e. spin-1 / . In theirscheme strong coupling between two ground states of anatomic ensemble is induced by balanced Raman transi-tions involving a cavity mode and a pump field. This ideacircumvents the thought to be unattainable condition forthe Dicke quantum phase transition which requires a cou-pling strength on the order of the energy separation be-tween the two involved atomic levels.In this work we realize the Dicke quantum phase transi-tion in an open system and observe self-organization of aBose-Einstein condensate. In the experiment, a conden-sate is trapped inside an ultrahigh-finesse optical cavity,and pumped from a direction transverse to the cavityaxis, as shown in figure 1. We will theoretically showthat the onset of self-organization is equivalent to theDicke quantum phase transition where the two-level sys-tem is formed by two different momentum states whichare coupled via the cavity field. At the phase transitiona spatial symmetry of the underlying lattice structure,given by the pump and cavity modes, is spontaneouslybroken. This steers the system from a flat superfluidphase into a quantum phase with macroscopic occupa-tion of the higher-order momentum mode and the cavitymode. The corresponding density wave together with thepresence of off-diagonal long-range order allows to regard a r X i v : . [ qu a n t - ph ] M a y the organized phase as a supersolid similar to thoseproposed for two-component systems . THEORETICAL DESCRIPTION AND THEDICKE MODEL
Let us first consider a single two-level atom of mass m interacting with a single cavity mode and the standing-wave pump field. The Hamiltonian then reads in aframe rotating with the pump laser frequencyˆ H (1) = ˆ p x + ˆ p z m + V cos ( k ˆ z ) + (cid:126) η (ˆ a † + ˆ a ) cos( k ˆ x ) cos( k ˆ z ) − (cid:126) (cid:16) ∆ c − U cos ( k ˆ x ) (cid:17) ˆ a † ˆ a. (1)Here, the excited atomic state is adiabatically eliminatedwhich is justified for large detuning ∆ a = ω p − ω a betweenthe pump laser frequency ω p and the atomic transitionfrequency ω a . The first term describes the kinetic en-ergy of the atom with momentum operators ˆ p x,z . Thepump laser creates a standing-wave potential of depth V = (cid:126) Ω p / ∆ a along the z -axis, where Ω p denotes themaximum pump Rabi frequency, and (cid:126) the Planck con-stant. Scattering between the pump field and the cav-ity mode, which is oriented along x , induces a latticepotential which dynamically depends on the scatteringrate and the relative phase between the pump field andthe cavity field. This phase is restricted to the values0 or π , for which the scattering induced light potentialhas a λ p / √ x - z direction, with λ p = 2 π/k denoting the pump wavelength (see Fig. 1c).The scattering rate is determined by the two-photon Rabifrequency η = g Ω p / ∆ a , with g being the atom-cavitycoupling strength. The last term describes the cavityfield, with photon creation and annihilation operators ˆ a † and ˆ a . The cavity resonance frequency ω c is detunedfrom the pump laser frequency by ∆ c = ω p − ω c , and thelight-shift of a single maximally coupled atom is given by U = g ∆ a .For a condensate of N atoms, the process ofself-organization can be captured by a mean-fielddescription . It assumes that all atoms occupy a sin-gle quantum state characterized by the wave function ψ , which is normalized to the atom number N . Thelight-atom interaction can now be described by a dy-namic light potential felt by all atoms. Since thetimescale of atomic dynamics in the motional degreeof freedom is much larger than the inverse of the cav-ity field decay rate κ , the coherent cavity field ampli-tude α adiabatically follows the atomic density distri-bution according to α = η Θ / (∆ c − U B + iκ ). Theorder parameter describing self-organization is given byΘ = (cid:104) ψ | cos( kx ) cos( kz ) | ψ (cid:105) which measures the localiza-tion of the atoms on either the even (Θ >
0) or the odd(Θ <
0) sublattice of the underlying checkerboard pat-tern defined by cos( kx ) cos( kz ) = ± abc even sitesodd sites P > P cr P < P cr λ p xyz FIG. 1. Concept of the experiment. A Bose-Einstein conden-sate which is placed inside an optical cavity is driven by astanding-wave pump laser oriented along the vertical z -axis.The frequency of the pump laser is far red-detuned with re-spect to the atomic transition line but close detuned to a par-ticular cavity mode. Correspondingly, the atoms coherentlyscatter pump light into the cavity mode with a phase depend-ing on their position within the combined pump–cavity modeprofile. a , For a homogeneous atomic density distributionalong the cavity axis, the build-up of a coherent cavity fieldis suppressed due to destructive interference of the individualscatterers. b , Above a critical pump power P cr the atomsself-organize onto either the even or odd sites of a checker-board pattern ( c ) thereby maximizing cooperative scatteringinto the cavity. This dynamical quantum phase transition istriggered by quantum fluctuations in the condensate density.It is accompanied by spontaneous symmetry breaking both inthe atomic density and the relative phase between pump fieldand cavity field. c , Geometry of the checkerboard pattern.The intensity maxima of the pump and cavity field are de-picted by the horizontal and vertical lines, respectively, with λ p denoting the pump wavelength. possible relative phases is adopted by the cavity field. Ac-cording to the spatial overlap between the atomic densityand the cavity mode profile, the atoms dispersively shiftthe cavity resonance proportional to B = (cid:104) ψ | cos ( kx ) | ψ (cid:105) .The resulting dynamic lattice potential reads V ( x, z ) = V cos ( kz ) + (cid:126) U | α | cos ( kx )+ (cid:126) η ( α + α ∗ ) cos( kx ) cos( kz ) . (2)The atoms self-organize due to positive feedback fromthe interference term in equation (2) above a criticaltwo-photon Rabi frequency η cr . Assuming that a densityfluctuation of the condensate induces e.g. Θ >
0, and thepump-cavity detuning is chosen to yield ∆ c − U B < . This analogy uses a two-mode description for the atomic field, where the initialBose-Einstein condensate is approximated by the zero-momentum state | p x , p z (cid:105) = | , (cid:105) . Photon scatteringbetween the pump and cavity field couples the zero-momentum state to the symmetric superposition of stateswhich carry an additional photon momentum along the x and z directions: |± (cid:126) k, ± (cid:126) k (cid:105) = (cid:80) µ,ν = ± | µ (cid:126) k, ν (cid:126) k (cid:105) / E r = (cid:126) ω r = (cid:126) k / (2 m ) compared to thezero-momentum state. (For the inclusion of Bloch states,see Methods.)There are two possible paths from the zero-momentumstate | p x , p z (cid:105) = | , (cid:105) to the excited momentum state | ± (cid:126) k, ± (cid:126) k (cid:105) : i) the absorption of a standing-wave pumpphoton followed by the emission into the cavity, ˆ a † ˆ J + ,and ii) the absorption of a cavity photon followed bythe emission into the pump field, ˆ a ˆ J + (see Fig. 2b).Here, the collective excitations to the higher-energy modeare expressed by the ladder operators ˆ J + = (cid:80) i | ± (cid:126) k, ± (cid:126) k (cid:105) i i (cid:104) , | = ˆ J †− , with the index i labelling theatoms. Including the reverse processes, the many-bodyinteraction Hamiltonian describing light scattering be-tween pump field and cavity field reads (see Methods) (cid:126) λ √ N (ˆ a † + ˆ a )( ˆ J + + ˆ J − ) . (3)This is exactly the interaction Hamiltonian of the Dickemodel which describes N two-level systems with transi-tion frequency ω interacting with a bosonic field mode atfrequency ω . This system exhibits a quantum phase tran-sition from a normal phase to a superradiant phase oncethe coupling strength λ between atoms and light reachesthe critical value of λ cr = √ ω ω/
2. Our system re-alizes the Dicke Hamiltonian with ω = − ∆ c + U N/ ab ± ¯ hk,0 | ± ¯ hk0, | ± ¯ hk, |± ¯ hk0,0 | ¯ hk, | ¯ hk∆ a g Ω p ω r Ω p g p x p z ˆa ˆJ + ˆa † ˆJ + ω a FIG. 2. Analogy to the Dicke model. In an atomic two-modepicture the pumped BEC–cavity system is equivalent to theDicke model including counter-rotating interaction terms. a ,Light scattering between the pump field and the cavity modeinduces two balanced Raman channels between the atomiczero-momentum state | p x , p z (cid:105) = | , (cid:105) and the symmetric su-perposition of the states | ± (cid:126) k, ± (cid:126) k (cid:105) with an additional pho-ton momentum along the x and z directions. b , The twoexcitation paths (dashed and solid) corresponding to the twoRaman channels are illustrated in a momentum diagram. Forthe notation see text. ω = 2 ω r and λ = η √ N /
2. Correspondingly, the pro-cess of self-organization is equivalent to the Dicke quan-tum phase transition where both the cavity field and theatomic polarization (cid:104) ˆ J + + ˆ J − (cid:105) = 2Θ acquire macroscopicoccupations.The experimental realization of the Dicke quantumphase transition is usually inhibited because the tran-sition frequencies by far exceed the available dipole cou-pling strengths. Using optical Raman transitions insteadbrings the energy difference between the atomic modesfrom the optical scale to a much lower energy scale,which makes the phase transition experimentally acces-sible. A similar realization of an effective Dicke Hamilto-nian has been theoretically considered using two balancedRaman channels between different electronic (instead ofmotional) states of an atomic ensemble interacting withan optical cavity and an external pump field . It is im-portant to point out that these systems are externallydriven and subject to cavity loss. Therefore they real-ize a dynamical version of the original Dicke quantumphase transition. However, the cavity output field offersthe unique possibility to in situ monitor the phase tran-sition as well as to extract important properties of thesystem . EXPERIMENTAL DESCRIPTION
Our experimental setup has been describedpreviously . In brief, we prepare almost pureBose-Einstein condensates of typically 10
Rb atomsin a crossed-beam dipole trap which is centered inside anultrahigh-finesse optical Fabry-Perot cavity. The atomsare prepared in the | F, m F (cid:105) = | , − (cid:105) hyperfine groundstate, where F denotes the total angular momentumand m F the magnetic quantum number. Perpendicularto the cavity axis the atoms are driven by a linearlypolarized standing-wave laser beam whose wavelength λ p is red-detuned by 4 . D line.The pump-atom detuning is more than five orders ofmagnitude larger than the atomic linewidth. Thisjustifies that we neglect spontaneous scattering in ourtheoretical description, and consider only coherent scat-tering between the pump beam and a particular TEM cavity mode which is quasi-resonant with the pump laserfrequency. The system operates in the regime of strongdispersive coupling where the maximum dispersiveshift of the empty cavity resonance induced by all atoms, N U , exceeds the cavity decay rate κ = 2 π × . . in-situ monitor the intracavity light inten-sity. In addition, we infer about the atomic momentumdistribution from absorption imaging along the y -axis af-ter a few milliseconds of free ballistic expansion of theatomic cloud. OBSERVING THE PHASE TRANSITION
To observe the onset of self-organization in thetransversally pumped BEC, we gradually increase thepump power over time while monitoring the light leakingout of the cavity, see figure 1. As long as the pump poweris kept below a threshold value no light is detected at thecavity output, and the expected momentum distributionof a condensate loaded into the shallow standing-wavepotential of the pump field is observed (see Fig. 3b,c).Once the pump power reaches the critical value an abruptbuild-up of the mean intracavity photon number marksthe onset of self-organization (see Fig. 3a). Simultane-ously, the atomic momentum distribution undergoes astriking change to show additional momentum compo-
FIG. 3. Observation of the phase transition. a , The pumppower (dashed) is gradually increased while monitoring themean intracavity photon number (solid, binned over 20 µ s).After sudden release and subsequent ballistic expansion of6 ms, absorption images (clipped equally in atomic density)are taken for different pump powers corresponding to latticedepths of: b , 2 . r , c , 7 . r , d , 8 . r . Self-organizationis manifested by an abrupt build-up of the cavity field ac-companied by the formation of momentum components at( p x , p z ) = ( ± (cid:126) k, ± (cid:126) k ) ( d ). The weak momentum componentsat ( p x , p z ) = (0 , ± (cid:126) k ) ( c ) originate from loading the atomsinto the 1D standing-wave potential of the pump laser. Thepump-cavity detuning was ∆ c = − π × . N = 1 . × . nents at ( p x , p z ) = ( ± (cid:126) k, ± (cid:126) k ) (see Fig. 3d). This pro-vides direct evidence for the acquired density modulationalong one of the two sublattices of a checkerboard patternassociated with a non-zero order parameter Θ.Conceptually, the self-organized quantum gas can beregarded as a supersolid , similar to those proposed fortwo-component systems . This requires the coexistenceof non-trivial diagonal long-range order correspondingto a periodic density modulation, and off-diagonal long-range order associated with phase coherence. In our sys-tem the checkerboard structure of the density modulationis determined by the long-range cavity-mediated atom-atom interactions in a non-trivial way. This is becausethe arrangement of the atoms is restricted to two possiblecheckerboard patterns which are intimately linked to thespontaneous breaking of the relative phase between pumpand cavity field. In contrast, the spatial atomic struc-ture in traditional optical lattice experiments is solely FIG. 4. Steady state in the self-organized phase. a , Pumppower sequence (dashed) and recorded mean intracavity pho-ton number (solid, binned over 20 µ s). After crossing thetransition point at 9 ms, the system reaches a steady statewithin the self-organized phase. The slow decrease in pho-ton number is due to atom loss (see text). The short-timefluctuations are due to detection shot-noise. b-d , Absorptionimages are taken after different times in the phase: ( b ) 3 ms,( c ) 7 ms, and ( d ) after lowering the pump power again to zero.The pump-cavity detuning was ∆ c = − π × . N = 0 . × . given by the externally applied light fields (see Meth-ods). In addition, the off-diagonal long-range order of theBose-Einstein condensate is not destroyed by the phasetransition. The atomic coherence length extends over al-most the full atomic ensemble, as we can deduce from thewidth of the higher-order momentum peaks in Fig. 3d.After crossing the phase transition the system quicklyreaches a steady state in the organized phase. As shownby a typical photon trace (see Fig. 4a), light is scatteredinto the cavity for up to 10 ms while the pump intensityis kept constant. This shows that the organized phase isstabilized by scattering induced light forces, which is instrict contrast to previous experiments observing (cavity-enhanced) superradiant light scattering where a nettransfer of momentum on the atomic cloud inhibited asteady state. The overall decrease of the mean cavityphoton number for constant pump intensity (see Fig. 4a)is attributed to atom loss caused by residual sponta-neous scattering at a rate of Γ sc = 3 . / s and backaction-induced heating of the atoms . Atom loss raises thecritical pump power according to P cr ∝ N − which, closeto the transition point, explains the observed reductionof the mean intracavity photon number. This was con-firmed by entering the organized phase twice within onerun and comparing the corresponding critical pump pow-ers of self-organization. From absorption imaging we de-duce an overall atom loss of 30% for the pump-powersequence shown in Fig. 4a. Experimentally however, theatom-loss induced photon-number reduction can be com-pensated for by either steadily increasing the pump in-tensity or chirping the pump-cavity detuning. From Fig. 4a we infer a maximum depth of the checker-board lattice potential of 22 E r which corresponds tosingle-site trapping frequencies of 19 kHz and 30 kHzalong the x - and z -direction, respectively. Accordingly,the atoms are confined to an array of tubes which are ori-ented along the weakly confined y -direction and containon average a few hundred atoms. Due to the stronglysuppressed tunnelling rate between adjacent tubes sep-arated by λ p / √ . This is directly observed via the reducedinterference contrast in the absorption images reflectingthat the supersolid phase evolved into a normal crys-talline phase (see Fig. 4b). However, the phase coherencebetween the tubes is quickly restored when the mean in-tracavity photon number decreases and the lattice depthcorrespondingly lowers (see Fig. 4c). After ramping thepump intensity to zero, an almost pure BEC is retrieved(see Fig. 4d). MAPPING OUT THE PHASE DIAGRAM
From the analogy to the Dicke quantum phase transi-tion we can deduce the dependence of the critical pumppower on the pump-cavity detuning ∆ c . To experimen-tally map out the phase boundary we gradually increasethe pump power similar to Fig. 3a for different values of∆ c (see Fig. 5b). The corresponding intracavity photonnumber traces are shown as a 2D color plot in Fig. 5a.A sharp phase boundary is observed over a wide rangeof pump-cavity detuning ∆ c . For large negative valuesof ∆ c the critical pump power P cr ∝ λ scales linearlywith the effective cavity frequency ω = − ∆ c + U B ,which agrees with the dependence expected from theDicke model (see Methods). For ω <
0, the critical cou-pling strength λ cr has no real solution. Indeed, almost nolight scattering is observed if the pump-cavity detuningis larger than the dispersively shifted cavity resonance at U B = − π × . B denotes the spatialoverlap between the cavity mode profile and the atomicdensity in the non-organized phase. As the pump-cavitydetuning approaches the shifted cavity resonance frombelow, scattering into the cavity and the intracavity pho-ton number increase.We quantitatively compare our measurements with thephase boundary calculated in a mean-field description,including the external confinement of the atoms, thetransverse pump and cavity mode profiles, and the colli-sional atom-atom interaction (see Methods). The agree-ment between measurements and theoretically expectedphase boundary is excellent (see Fig. 5a, dashed curve).The organization of the atoms on a checkerboard pat-tern not only affects the scattering rate between pumpand cavity field, but also changes the spatial overlap B . This dynamically shifts the cavity resonance, whichgoes beyond the Dicke model (see Methods), and resultsin a frustrated system for U N > ∆ c > U B . Herethe onset of self-organization brings the coupled atoms- FIG. 5. Phase diagram. a , The pump power is increased to 1 . c . The recorded mean intracavity photon number ¯ n is displayed in color along the rescaled horizontal axis, showing pumppower and corresponding pump lattice depth. A sharp phase boundary is observed over a wide range of the pump-cavitydetuning ∆ c , which is in very good agreement with a theoretical mean-field model (dashed curve). The dispersively shiftedcavity resonance for the non-organized atom cloud is marked by the arrow. b-c , Typical traces showing the intracavity photonnumber for different pump-cavity detuning: ( b ) ∆ c = − π × . µ s, ( c ) ∆ c = − π × . µ s. The atom number was N = 1 . × . In the detuning range − π × ≥ ∆ c ≥ − π ×
21 MHz thepump power ramp was interrupted at 540 µ W. Therefore, no photon data was taken under the insets. cavity system into resonance with the pump laser, andthe positive feedback which drives self-organization is in-terrupted (see Eq. 2). Experimentally this is observed inan oscillatory behavior of the system between the orga-nized and the non-organized phase (see Fig. 5c).
CONCLUSIONS AND OUTLOOK
We have experimentally realized a second-order dy-namical quantum phase transition in a driven Bose-Einstein condensate coupled to the field of an ultrahigh-finesse optical cavity. At a critical driving strengththe steady state realized by the system spontaneouslybreaks an Ising-type symmetry accompanied by self-organization of the superfluid atoms. We identify regimeswhere the emergent light-atom crystal is accompanied byphase coherence, and can thus be considered as a super-solid. The process of self-organization is shown to beequivalent to the Dicke quantum phase transition in anopen system. We gain experimental access to the phasediagram of the Dicke model by observing the cavity out-put in situ . In a very cold classical gas the correspondingphase boundary is predicted to scale with the tempera-ture instead of the recoil energy , and the transition isdriven by classical fluctuations in the atomic density in- stead of quantum fluctuations.For the presented experiments the collective interac-tion λ cr between the induced atomic dipoles and the cav-ity field approaches the order of the cavity decay rate κ , with a maximum ratio of λ cr /κ = 0 .
2. Reaching theregime where the Hamiltonian dynamics dominates thecavity losses offers possibilities to study the coherent dy-namics of the Dicke model at the critical point whichwas shown theoretically to be dominated by macroscopicatom-field and atom-atom entanglement . Detect-ing the phase of the light leaving the resonator opensthe opportunity to study spontaneous symmetry break-ing induced by pure quantum fluctuations. Furthermore,recording the statistics of the scattered light may enablequantum non-demolition measurements and the prepara-tion of exotic many-body states . METHODSEXPERIMENTAL DETAILS
We prepare almost pure Rb Bose-Einstein conden-sates in a crossed-beam dipole trap with trapping fre-quencies of ( ω x , ω y , ω z ) = 2 π × (252 , , x denotes the cavity axis and z the pump axis. For a typ-ical atom number of N = 10 this results in condensateradii of ( R x , R y , R z ) = (3 . , . , . µ m which were de-duced in a mean-field approximation . Experimentally,the position of the dipole trap is aligned to maximize thespatial overlap between the BEC and the cavity TEM mode which has a waist radius of w c = 25 µ m. The cavityhas a finesse of 3 . × . Its length of 178 µ m is activelystabilized using a weak laser beam at 830 nm which is ref-erenced onto the transverse pump laser . The intracav-ity stabilization light results in a weak lattice potentialwith a depth of less than 0 .
35 E r .The pump laser beam has waist radii of ( w x , w y ) =(29 , µ m at the position of the atoms. To accomplishoptimal mode matching with the atomic cloud we usethe same optical fiber for the pump light and the ver-tical beam of the crossed-beam dipole trap. The retro-reflected pump power is reduced by a factor of 0 . λ p = 784 . y -axis (within an uncertaintyof 5 %) to optimize scattering into the cavity mode. Aweak magnetic field of 0 . | F, m F (cid:105) = | , − (cid:105) ground state. Accordingly, only σ + or σ − polarized photons can be scattered into thecavity mode. We observe the onset of self-organizationalways with σ + polarized cavity light since the corre-sponding atom-cavity coupling strength exceeds the onefor σ − polarized light.The light which leaks out of the cavity is monitored ontwo single-photon counting modules each of which is sen-sitive to one of the two different circular polarizations. Inprinciple this allows to detect single intracavity photonswith an efficiency of about 5%. However, for the experi-ments reported in this work the detection efficiency wasreduced by a factor of 10 in order to enlarge the dynami-cal range of our light detection (limited by the saturationeffects of the photon counting modules). The systematicuncertainties in determining the intracavity photon num-ber is estimated to be 25 %. MAPPING TO THE DICKE HAMILTONIAN
The onset of self-organization is equivalent to a dynam-ical version of the normal to superradiant quantum phase transition of the Dicke model. This analogy is derived ina two-mode expansion of the atomic matter field, and al-lows to directly infer about properties of the transitioninto the organized phase. A one-dimensional analysis ofthe mapping to the Dicke model has been developed in-dependently in Ref. .In the absence of collisional atom-atom interactionsthe many-body Hamiltonian describing the driven BEC–cavity system is given byˆ H = (cid:90) ˆΨ † ( x, z ) ˆ H (1) ˆΨ( x, z )d x d z (4)where ˆΨ denotes the atomic field operator, and ˆ H (1) is the single-particle Hamiltonian given in equation (1).In the non-organized phase the mean intracavity pho-ton number vanishes and all atoms occupy the lowest-energy Bloch state ψ of the 1D lattice Hamiltonian ˆ p z m + V cos ( k ˆ z ). Scattering of photons between thepump field and the cavity mode couples the state ψ to the state ψ ∝ ψ cos( kx ) cos( kz ) which carries addi-tional (cid:126) k momentum components along the x and z direc-tion. In order to understand the onset of self-organizationwe expand the field operator ˆΨ in the reduced Hilbertspace spanned by the modes ψ and ψ . Note that, fordescribing the deeply organized phase, higher-order mo-mentum states have to be included in the description inorder to account for atomic localization at the sites ofthe emergent checkerboard pattern.After inserting the expansion ˆΨ = ψ ˆ c + ψ ˆ c into themany-body Hamiltonian (see Eq. 4) we obtain up to aconstant termˆ H/ (cid:126) = ω ˆ J z + ω ˆ a † ˆ a + λ √ N (ˆ a † +ˆ a )( ˆ J + + ˆ J − )+ 34 U ˆ c † ˆ c ˆ a † ˆ a, (5)with bosonic mode operators ˆ c and ˆ c , and the totalatom number N = ˆ c † ˆ c + ˆ c † ˆ c . Here, the collective spinoperators ˆ J + = ˆ c † ˆ c = ˆ J †− and ˆ J z = (ˆ c † ˆ c − ˆ c † ˆ c )were introduced. Apart from the last term, ˆ H is theDicke Hamiltonian which describes the coupling be-tween N two-level systems with transition frequency ω = 2 ω r and a bosonic field mode with frequency ω = − ∆ c + N U /
2. Their collective coupling strengthis given by λ = √ N η/
2, which experimentally can betuned by varying the pump laser power. The last termin equation (5) describes the dynamic (dispersive) shift ofthe cavity frequency, which is negligible in the close vicin-ity of the phase transition. Therefore, self-organizationof the transversally pumped BEC–cavity system corre-sponds to the quantum phase transition of the Dickemodel from a normal into a superradiant phase .The Dicke Hamiltonian is invariant under the paritytransformation ˆ a → − ˆ a and ˆ J ± → − ˆ J ± . This sym-metry is spontaneously broken by the process of self-organization corresponding to the atomic arrangementon the even or odd sites of a checkerboard pattern with (cid:104) ˆ J + + ˆ J − (cid:105) = 2Θ taking either positive or negative val-ues. At the same time the relative phase between thepump and cavity field takes one of two possible valuesseparated by π . This is in contrast to traditional opti-cal lattice experiments where the phase between differentlaser beams determining their interference pattern is ex-ternally controlled . DERIVATION OF THE PHASE BOUNDARY INA MEAN-FIELD DESCRIPTION
To derive a quantitative expression for the criticalpump intensity of self-organization, we perform a sta-bility analysis of the compound BEC–cavity system in amean-field description, following Ref. . For comparisonwith our experimental findings we take into account theexternal trapping potential, the transverse sizes of thecavity mode and the pump beam, as well as collisionalatom-atom interactions. The system is described by thegeneralized Gross-Pitaevskii equation (cid:16) p m + V ext ( r ) + (cid:126) U | α | φ c ( r ) + (cid:126) η ( α + α ∗ ) φ c ( r ) φ p ( r )+ g | ψ | (cid:17) ψ ( r , t ) = µψ ( r , t ) (6)where ψ ( r ) denotes the condensate wave function (nor-malized to the total atom number N ), and α denotes thecoherent cavity field amplitude which was adiabaticallyeliminated according to: α = η Θ∆ c − U B + iκ . (7)The mode profiles of the cavity and the pump beamare given by φ c ( r ) = cos( kx ) e − y z w c and φ p ( r ) =cos( kz ) e − x w x − y w y , respectively. The external poten-tial V ext consists of the harmonic trapping potential m ( ω x x + ω y y + ω z z ) / V φ p ( r ) provided bythe pump beam. The order parameter Θ = (cid:104) ψ | φ c φ p | ψ (cid:105) and the bunching parameter B = (cid:104) ψ | φ c | ψ (cid:105) are definedaccording to the main text. The collisional interactionstrength is given by g = π (cid:126) am with the s-wave scatter- ing length a . The chemical potential of the condensateis denoted by µ .A defining condition for the critical two-photon Rabifrequency η cr is obtained from a linear stability anal-ysis of equation (6) around the non-organized phase ψ with α = 0. Starting with the two-mode ansatz ψ = ψ (1 + (cid:15)φ c φ p ) with (cid:15) (cid:28)
1, we carry out an infinites-imal propagation step into imaginary time in equation(6). This yields the following condition for the criticalpump strength η cr where the system exhibits a dynami-cal instability η cr (cid:112) N eff = 12 (cid:115) ˜∆ c + κ − ˜∆ c (cid:112) ω r + 4 E int / (cid:126) . (8)Here, we introduced the effective number of maximallyscattering atoms N eff = (cid:104) ψ | φ c φ p | ψ (cid:105) , and denoted thedetuning between the pump frequency and the disper-sively shifted cavity resonance by ˜∆ c = ∆ c − U B , with B = (cid:104) ψ | φ c | ψ (cid:105) . The interaction energy per particle,given by E int = g N (cid:82) | ψ | d r , accounts for the mean-fieldshift of the free-particle dispersion relation.Identifying ω = − ˜∆ c , ω = 2 ω r + 4 E int / (cid:126) and λ cr = η cr √ N eff our result agrees with the critical cou-pling strength λ cr obtained in the Dicke model includingcavity decay λ cr = 12 (cid:114) ω + κ ω ω . (9)The phase boundary shown in Fig. 5a (dashed curve)is obtained from equation (8) by approximating the con-densate wave function ψ by the Thomas-Fermi solutionin the crossed-beam dipole trap . ACKNOWLEDGMENTS
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